,TESTBANK For Elementary Linear Algebra with
Applications 12e Anton
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, Chapter 1: Systems of Linear Equations and Matrices
Multiple Choice Questions
1. Which of the following equations is linear?
(A) 2x21 + 3x32 + 4x43 = 5
√ √
(B) 3x1 − 2x2 + x3 = 5
√ √
(C) 5x1 + 5 x2 − x3 = 1
(D) 22 x1 + cos (x2 ) + 4x3 = 7
2. Which system corresponds to the following augmented matrix?
1 11 6 3
9 4 0 −2
x1 + 11x2 = −3
(A)
9x1 + 4x2 = −2
x1 + 11x2 + 6x3 = 3
(B)
9x1 + 4x2 = −2
x1 + 11x2 + 6x3 + 3x4 = 0
(C)
9x1 + 4x2 − 2x4 = 0
x1 + 9x2 = 0
11x1 + 4x2 = 0
(D)
6x1 =0
3x1 − 2x2 = 0
3. Which of the following statements best describes the following augmented matrix?
⎡ ⎤
1 2 6 5
⎢ ⎥
⎢
A = ⎣−1 1 −2 3⎥ ⎦
1 −4 −2 1
(A) A is consistent with a unique solution.
(B) A is consistent with infinitely many solutions.
(C) A is inconsistent.
(D) none of the above.
,Elementary Linear Algebra 12e –2– Anton/Rorres
4. Which of the following matrices is in reduced row echelon form?
⎡ ⎤
1 0 −1 1
⎢ ⎥
(A) ⎢
⎣0 1 2 0⎥ ⎦
0 1 3 1
⎡ ⎤
1 0 2 5
⎢ ⎥
⎢
(B) ⎣0 1 −7 5⎥⎦
0 0 1 14
1 0 0 11 −3
(C)
0 0 0 1 4
⎡ ⎤
1 0 −5
⎢ ⎥
(D) ⎢
⎣0 1 3⎥
⎦
0 0 0
5. If the matrix A is 4 × 2, B is 3 × 4, C is 2 × 4, D is 4 × 3, and E is 2 × 5, which of the
following expressions is not defined?
(A) AT D + CB T (B) (B + DT )A (C) CA + CB T (D) DBAE
6. What is the second row of the product AB?
⎡ ⎤ ⎡ ⎤
0 2 3 2 1 7
⎢ ⎥ ⎢ ⎥
A=⎢ ⎥ ⎢
⎣5 4 8 ⎦ , B = ⎣ 6 3 2⎥
⎦
9 7 2 2 9 7
(A) 18 33 25 (B) 64 48 91 (C) 50 89 99 (D) 48 89 33
a b
7. Which of the following is the determinant of the 2 × 2 matrix A = ?
c d
1 1
(A) ad − bc (B) bc − ad (C) bc−ad
(D) ad−bc
8. Which of the following matrices is not invertible?
3 6 7 7 9 0 9 3
(A) (B) (C) (D)
2 4 2 3 4 4 6 5
9. Which of the following matrices is not an elementary matrix?
⎡ ⎤ ⎡ ⎤
0 1 0 1 0 0
1 0 1 1 ⎢ ⎥ ⎢ ⎥
(A) (B) (C) ⎢ ⎥
⎣1 0 0⎦ (D) ⎣0 −2
⎢ 0⎥
⎦
5 1 0 2
0 0 1 0 0 1
Chapter 1
,Elementary Linear Algebra 12e –3– Anton/Rorres
10. For which elementary matrix E will the equation EA = B hold?
⎡ ⎤ ⎡ ⎤
1 4 6 1 4 6
⎢ ⎥ ⎢ ⎥
A=⎢ ⎣0 0 1⎥⎦ , B = ⎢0 0
⎣ 1 ⎥
⎦
2 10 9 0 2 −3
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
1 0 0 1 0 0 1 0 0 0 0 1
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
(A) ⎢
⎣ 0 1 0 ⎥ (B) ⎢0 0 1⎥
⎦ ⎣ ⎦ (C) ⎢ ⎥
⎣ 0 1 0⎦ (D) ⎣0
⎢ 1 0⎥
⎦
2 0 1 0 1 0 −2 0 1 1 0 0
11. Which matrix will be used as the inverted coefficient matrix when solving the following
system?
3x1 + x2 = 4
5x1 + 2x2 = 7
2 −1 −2 1 2 1 −2 −1
(A) (B) (C) (D)
−5 3 5 −3 5 3 −5 −3
12. What value of b makes the following system consistent?
4x1 + 2x2 = b
2x1 + x2 = 0
(A) b = −1 (B) b = 0 (C) b = 1 (D) b = 2
13. If A is a 3 × 3 diagonal matrix, which of the following matrices is not a possible value
of Ak for some integer k?
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
0 0 0 1 0 1 1 0 0 0 0 0
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
(A) ⎢ ⎥
⎣0 4 0⎦ (B) ⎣0 16
⎢ 0⎥ ⎢
⎦ (C) ⎣0 4
1
0⎥⎦ (D) ⎣0
⎢ 0 0⎥
⎦
0 0 9 4 0 25 0 0 −1 0 0 0
⎡ ⎤
3 0 0
⎢ ⎥
14. The matrix ⎢ ⎣0 −7 0⎦ is:
⎥
0 0 1
(A) upper triangular.
(B) lower triangular.
(C) both (A) and (B).
(D) neither (A) nor (B).
Chapter 1
,Elementary Linear Algebra 12e –4– Anton/Rorres
15. If A is a 4×5 matrix, find the domain and codomain of the transformation TA (x) = Ax.
(A) Not enough information
(B) Domain: R4 , Codomain: R5
(C) Domain: R5 , Codomain: R5
(D) Domain: R5 , Codomain: R4
16. Which of the following is a matrix transformation?
(A) T (x, y, z) = (yx2 , yz 2 )
(B) T (x, y, z, w) = (xy, yz, zw, wx)
(C) T (x, y, z) = (x + 1, x + 2, x + z, y + z)
(D) T (x, y) = (4x, 5x, −x, 0)
17. Which matrix represents reflection about the xy-plane?
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
−1 0 0 1 0 0 1 0 0 −1 0 0
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
(A) ⎢
⎣ 0 1 0 ⎥ (B) ⎢0 −1
⎦ ⎣ 0 ⎥ (C) ⎢0 1
⎦ ⎣ 0 ⎥
⎦ (D) ⎢
⎣ 0 −1 0⎥
⎦
0 0 1 0 0 −1 0 0 −1 0 0 1
18. Use matrix multiplication to find the image of the vector 2, 1 when it is rotated
counterclockwise about the origin through an angle θ = 45◦ .
√ √ √ √ √ √ √ √
2
(A) 2 , 2 3 2 3 2
(B) 2 , 2 2 2
(C) − 2 , 23 2 (D) − 3 2 2 , 22
19. Which of the following pairs of operators T1 , T2 : R2 → R2 commute? (That is, for
which pair is it true that T1 ◦ T2 = T2 ◦ T1 ?)
(A) T1 is the reflection about the x-axis.
T2 is the reflection about line y = x.
(B) T1 is the orthogonal projection onto the x-axis.
T2 is the reflection about line y = x.
(C) T1 is the counterclockwise rotation about the origin through an angle of π.
T2 is the projection onto the y-axis.
(D) T1 is the reflection about the x-axis.
T2 is the counterclockwise rotation about the origin through an angle of π/2.
Free Response Questions
1. Find the relationship between a and b such that the following system has infinitely many
solutions.
−x + 2y = a
−3x + 6y = b
Chapter 1
,Elementary Linear Algebra 12e –5– Anton/Rorres
2. Solve the following system and use parametric equations to describe the solution set.
x1 + 2x2 + 3x3 = 11
2x1 − x2 + x3 = 2
3x1 + x2 + 4x3 = 13
3. Determine whether the following system has no solution, exactly one solution, or infinitely
many solutions.
2x1 + 2x2 = 2
x1 + x 2 = 4
15 −3 6
4. Find the value of k that makes the system inconsistent.
−10 k 9
5. Solve the following system using Gaussian elimination.
x1 − x2 − 5x3 = −1
−2x1 + 2x2 + 11x3 = 1
3x1 − x2 + x3 = 3
6. Solve the following system for x, y, and z.
1 1 1
x
− y
− z
=0
2 1 1
x
+ y
+ z
=3
3 1
x
− z
=0
7. The curve y = ax3 + bx2 + x + c passes through the points (0, 0), (1, 1), and (−1, −2).
Find and solve a system of linear equations to determine the values of a, b, and c.
8. Solve the following system for x and y.
x2 + y 2 = 6
x2 − y 2 = 2
1 −1
9. Given C = , find CC T .
2 0
10. Express the following matrix equation as a system of linear equations.
⎡ ⎤⎡ ⎤ ⎡ ⎤
−1 7 0 x 0
⎢ ⎥⎢ ⎥ ⎢ ⎥
⎢ 0 4 3⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ y ⎦ = ⎣0 ⎦
6 0 −2 z 0
Chapter 1
,Elementary Linear Algebra 12e –6– Anton/Rorres
11. Find the 3 × 3 matrix A = [aij ] whose entries satisfy the condition aij = i2 − j.
12. Let A and B be n × n matrices. Prove that tr (c · A − B) = c · tr (A) − tr (B).
4 0
13. What is the inverse of ?
9 2
4 4
14. Given the polynomial p(x) = x2 − 3x + 1 and the matrix A = , compute p(A).
6 1
15. Let A, B, C, and D be n × n invertible matrices. Solve for A given that the following
equation holds.
C 2 DA−1 CB −1 = BCB −1
16. Prove that for any m × n matrices A and B, (A − B)T = AT − B T .
17. Use the inversion algorithm to find the inverse of the following matrix.
⎡ ⎤
1 2 1
⎢ ⎥
⎢0 2 2⎥
⎣ ⎦
0 0 4
18. Which elementary row operation will transform the following matrix into the identity
matrix? ⎡ ⎤
1 0 0 0
⎢ ⎥
⎢0 1 0 0⎥
⎢ ⎥
⎢ ⎥
⎢0 0 1 0 ⎥
⎣ ⎦
0 −9 0 1
19. Find the 3 × 3 elementary matrix that adds c times row 3 to row 1.
20. Find the elementary matrix E that satisfies
⎡ ⎤ ⎡ ⎤
1 4 6 1 4 6
⎢ ⎥ ⎢ ⎥
⎢
E ⎣0 ⎥
0 1⎦ = ⎣ 0⎢ 0 1⎥
⎦
2 10 9 0 2 −3
21. Solve the following system by inverting the coefficient matrix.
7x + 2y = 1
3x + y = 5
Chapter 1
,Elementary Linear Algebra 12e –7– Anton/Rorres
22. Solve the following matrix equation for X.
⎡ ⎤ ⎡ ⎤
1 2 3 2 2 3 0
⎢ ⎥ ⎢ ⎥
⎢0 1 4⎥ X = ⎢ 0 0 0 1⎥
⎣ ⎦ ⎣ ⎦
5 6 0 3 1 1 1
⎡ ⎤ ⎡ ⎤
1 0 0 3
⎢ ⎥ ⎢ ⎥
23. Given that A = ⎢
−1 ⎥
⎣0 2 1⎦ and b = ⎣1⎦,
⎢ ⎥ solve the system A2 x = b.
0 1 0 2
24. Find a nonzero solution to the following equation.
1 3
x = 3x
4 −3
25. Find the values of a, b, and c that make the following matrix symmetric.
⎡ ⎤
3 a 2−b
⎢ ⎥
⎢ 4 0 a + b⎥
⎣ ⎦
2 c 7
⎡ ⎤ ⎡ ⎤
3 4 3 1 −7 6
⎢ ⎥ ⎢ ⎥
26. Let A = ⎢
⎣ 0 0 6 ⎥, B = ⎢−4
⎦ ⎣ 5 0 ⎥, and AB = [cij ].
⎦
0 0 2 1 0 2
Find the diagonal entries c11 , c22 , and c33 .
27. Let the entries of a matrix A = [aij ] be defined as aij = 2i2 − i + j + g(j), where g is a
function of j. If A is a symmetric matrix, what is g(j)?
28. Prove that for any square matrix A, the matrix B = (A + AT ) is symmetric.
29. Find the domain and codomain of the transformation defined by
⎡ ⎤
x1
⎢ ⎥
5 7 6 0 ⎢ x2 ⎥
⎢
⎥
⎢ ⎥
1 0 −2 −2 ⎢ ⎥
⎣ x3 ⎦
x4
30. Find the standard matrix for the operator T : R2 → R2 defined by
3x1 + x2 = w1
4x2 = w2
Chapter 1
, Elementary Linear Algebra 12e –8– Anton/Rorres
31. Find the standard matrix for the transformation T defined by the formula
T (x1 , x2 , x3 ) = (x1 , −x3 , x2 − x1 , 3x2 + x3 )
32. Find the standard matrix A for the linear transformation T : R2 → R2 for which
2 3 −1 5
T = ,T =
−1 −4 1 2
33. Prove that the composition of two rotation operators about the origin of R2 is another
rotation about the origin.
34. Prove that if TA : R3 → R3 and TA (x) = 0 for every vector x in R3 , then A is the 3 × 3
zero matrix.
35. Write a balanced equation for the following chemical reaction.
C3 H8 + O2 → H2 O + CO2
36. Find the quadratic polynomial whose graph passes through the points 0, 3 , −1, 8 ,
and 1, 0 .
37. Use matrix inversion to find the production vector x that meets the demand d for the
consumption matrix C. ⎡ ⎤ ⎡ ⎤
0.1 0.3 0.2 18
⎢ ⎥ ⎢ ⎥
C=⎢ ⎥ ⎢ ⎥
⎣0.5 0.1 0.2⎦ ; d = ⎣40⎦
0.2 0.4 0.3 26
Chapter 1
Applications 12e Anton
Hello all ,
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, Chapter 1: Systems of Linear Equations and Matrices
Multiple Choice Questions
1. Which of the following equations is linear?
(A) 2x21 + 3x32 + 4x43 = 5
√ √
(B) 3x1 − 2x2 + x3 = 5
√ √
(C) 5x1 + 5 x2 − x3 = 1
(D) 22 x1 + cos (x2 ) + 4x3 = 7
2. Which system corresponds to the following augmented matrix?
1 11 6 3
9 4 0 −2
x1 + 11x2 = −3
(A)
9x1 + 4x2 = −2
x1 + 11x2 + 6x3 = 3
(B)
9x1 + 4x2 = −2
x1 + 11x2 + 6x3 + 3x4 = 0
(C)
9x1 + 4x2 − 2x4 = 0
x1 + 9x2 = 0
11x1 + 4x2 = 0
(D)
6x1 =0
3x1 − 2x2 = 0
3. Which of the following statements best describes the following augmented matrix?
⎡ ⎤
1 2 6 5
⎢ ⎥
⎢
A = ⎣−1 1 −2 3⎥ ⎦
1 −4 −2 1
(A) A is consistent with a unique solution.
(B) A is consistent with infinitely many solutions.
(C) A is inconsistent.
(D) none of the above.
,Elementary Linear Algebra 12e –2– Anton/Rorres
4. Which of the following matrices is in reduced row echelon form?
⎡ ⎤
1 0 −1 1
⎢ ⎥
(A) ⎢
⎣0 1 2 0⎥ ⎦
0 1 3 1
⎡ ⎤
1 0 2 5
⎢ ⎥
⎢
(B) ⎣0 1 −7 5⎥⎦
0 0 1 14
1 0 0 11 −3
(C)
0 0 0 1 4
⎡ ⎤
1 0 −5
⎢ ⎥
(D) ⎢
⎣0 1 3⎥
⎦
0 0 0
5. If the matrix A is 4 × 2, B is 3 × 4, C is 2 × 4, D is 4 × 3, and E is 2 × 5, which of the
following expressions is not defined?
(A) AT D + CB T (B) (B + DT )A (C) CA + CB T (D) DBAE
6. What is the second row of the product AB?
⎡ ⎤ ⎡ ⎤
0 2 3 2 1 7
⎢ ⎥ ⎢ ⎥
A=⎢ ⎥ ⎢
⎣5 4 8 ⎦ , B = ⎣ 6 3 2⎥
⎦
9 7 2 2 9 7
(A) 18 33 25 (B) 64 48 91 (C) 50 89 99 (D) 48 89 33
a b
7. Which of the following is the determinant of the 2 × 2 matrix A = ?
c d
1 1
(A) ad − bc (B) bc − ad (C) bc−ad
(D) ad−bc
8. Which of the following matrices is not invertible?
3 6 7 7 9 0 9 3
(A) (B) (C) (D)
2 4 2 3 4 4 6 5
9. Which of the following matrices is not an elementary matrix?
⎡ ⎤ ⎡ ⎤
0 1 0 1 0 0
1 0 1 1 ⎢ ⎥ ⎢ ⎥
(A) (B) (C) ⎢ ⎥
⎣1 0 0⎦ (D) ⎣0 −2
⎢ 0⎥
⎦
5 1 0 2
0 0 1 0 0 1
Chapter 1
,Elementary Linear Algebra 12e –3– Anton/Rorres
10. For which elementary matrix E will the equation EA = B hold?
⎡ ⎤ ⎡ ⎤
1 4 6 1 4 6
⎢ ⎥ ⎢ ⎥
A=⎢ ⎣0 0 1⎥⎦ , B = ⎢0 0
⎣ 1 ⎥
⎦
2 10 9 0 2 −3
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
1 0 0 1 0 0 1 0 0 0 0 1
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
(A) ⎢
⎣ 0 1 0 ⎥ (B) ⎢0 0 1⎥
⎦ ⎣ ⎦ (C) ⎢ ⎥
⎣ 0 1 0⎦ (D) ⎣0
⎢ 1 0⎥
⎦
2 0 1 0 1 0 −2 0 1 1 0 0
11. Which matrix will be used as the inverted coefficient matrix when solving the following
system?
3x1 + x2 = 4
5x1 + 2x2 = 7
2 −1 −2 1 2 1 −2 −1
(A) (B) (C) (D)
−5 3 5 −3 5 3 −5 −3
12. What value of b makes the following system consistent?
4x1 + 2x2 = b
2x1 + x2 = 0
(A) b = −1 (B) b = 0 (C) b = 1 (D) b = 2
13. If A is a 3 × 3 diagonal matrix, which of the following matrices is not a possible value
of Ak for some integer k?
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
0 0 0 1 0 1 1 0 0 0 0 0
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
(A) ⎢ ⎥
⎣0 4 0⎦ (B) ⎣0 16
⎢ 0⎥ ⎢
⎦ (C) ⎣0 4
1
0⎥⎦ (D) ⎣0
⎢ 0 0⎥
⎦
0 0 9 4 0 25 0 0 −1 0 0 0
⎡ ⎤
3 0 0
⎢ ⎥
14. The matrix ⎢ ⎣0 −7 0⎦ is:
⎥
0 0 1
(A) upper triangular.
(B) lower triangular.
(C) both (A) and (B).
(D) neither (A) nor (B).
Chapter 1
,Elementary Linear Algebra 12e –4– Anton/Rorres
15. If A is a 4×5 matrix, find the domain and codomain of the transformation TA (x) = Ax.
(A) Not enough information
(B) Domain: R4 , Codomain: R5
(C) Domain: R5 , Codomain: R5
(D) Domain: R5 , Codomain: R4
16. Which of the following is a matrix transformation?
(A) T (x, y, z) = (yx2 , yz 2 )
(B) T (x, y, z, w) = (xy, yz, zw, wx)
(C) T (x, y, z) = (x + 1, x + 2, x + z, y + z)
(D) T (x, y) = (4x, 5x, −x, 0)
17. Which matrix represents reflection about the xy-plane?
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
−1 0 0 1 0 0 1 0 0 −1 0 0
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
(A) ⎢
⎣ 0 1 0 ⎥ (B) ⎢0 −1
⎦ ⎣ 0 ⎥ (C) ⎢0 1
⎦ ⎣ 0 ⎥
⎦ (D) ⎢
⎣ 0 −1 0⎥
⎦
0 0 1 0 0 −1 0 0 −1 0 0 1
18. Use matrix multiplication to find the image of the vector 2, 1 when it is rotated
counterclockwise about the origin through an angle θ = 45◦ .
√ √ √ √ √ √ √ √
2
(A) 2 , 2 3 2 3 2
(B) 2 , 2 2 2
(C) − 2 , 23 2 (D) − 3 2 2 , 22
19. Which of the following pairs of operators T1 , T2 : R2 → R2 commute? (That is, for
which pair is it true that T1 ◦ T2 = T2 ◦ T1 ?)
(A) T1 is the reflection about the x-axis.
T2 is the reflection about line y = x.
(B) T1 is the orthogonal projection onto the x-axis.
T2 is the reflection about line y = x.
(C) T1 is the counterclockwise rotation about the origin through an angle of π.
T2 is the projection onto the y-axis.
(D) T1 is the reflection about the x-axis.
T2 is the counterclockwise rotation about the origin through an angle of π/2.
Free Response Questions
1. Find the relationship between a and b such that the following system has infinitely many
solutions.
−x + 2y = a
−3x + 6y = b
Chapter 1
,Elementary Linear Algebra 12e –5– Anton/Rorres
2. Solve the following system and use parametric equations to describe the solution set.
x1 + 2x2 + 3x3 = 11
2x1 − x2 + x3 = 2
3x1 + x2 + 4x3 = 13
3. Determine whether the following system has no solution, exactly one solution, or infinitely
many solutions.
2x1 + 2x2 = 2
x1 + x 2 = 4
15 −3 6
4. Find the value of k that makes the system inconsistent.
−10 k 9
5. Solve the following system using Gaussian elimination.
x1 − x2 − 5x3 = −1
−2x1 + 2x2 + 11x3 = 1
3x1 − x2 + x3 = 3
6. Solve the following system for x, y, and z.
1 1 1
x
− y
− z
=0
2 1 1
x
+ y
+ z
=3
3 1
x
− z
=0
7. The curve y = ax3 + bx2 + x + c passes through the points (0, 0), (1, 1), and (−1, −2).
Find and solve a system of linear equations to determine the values of a, b, and c.
8. Solve the following system for x and y.
x2 + y 2 = 6
x2 − y 2 = 2
1 −1
9. Given C = , find CC T .
2 0
10. Express the following matrix equation as a system of linear equations.
⎡ ⎤⎡ ⎤ ⎡ ⎤
−1 7 0 x 0
⎢ ⎥⎢ ⎥ ⎢ ⎥
⎢ 0 4 3⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ y ⎦ = ⎣0 ⎦
6 0 −2 z 0
Chapter 1
,Elementary Linear Algebra 12e –6– Anton/Rorres
11. Find the 3 × 3 matrix A = [aij ] whose entries satisfy the condition aij = i2 − j.
12. Let A and B be n × n matrices. Prove that tr (c · A − B) = c · tr (A) − tr (B).
4 0
13. What is the inverse of ?
9 2
4 4
14. Given the polynomial p(x) = x2 − 3x + 1 and the matrix A = , compute p(A).
6 1
15. Let A, B, C, and D be n × n invertible matrices. Solve for A given that the following
equation holds.
C 2 DA−1 CB −1 = BCB −1
16. Prove that for any m × n matrices A and B, (A − B)T = AT − B T .
17. Use the inversion algorithm to find the inverse of the following matrix.
⎡ ⎤
1 2 1
⎢ ⎥
⎢0 2 2⎥
⎣ ⎦
0 0 4
18. Which elementary row operation will transform the following matrix into the identity
matrix? ⎡ ⎤
1 0 0 0
⎢ ⎥
⎢0 1 0 0⎥
⎢ ⎥
⎢ ⎥
⎢0 0 1 0 ⎥
⎣ ⎦
0 −9 0 1
19. Find the 3 × 3 elementary matrix that adds c times row 3 to row 1.
20. Find the elementary matrix E that satisfies
⎡ ⎤ ⎡ ⎤
1 4 6 1 4 6
⎢ ⎥ ⎢ ⎥
⎢
E ⎣0 ⎥
0 1⎦ = ⎣ 0⎢ 0 1⎥
⎦
2 10 9 0 2 −3
21. Solve the following system by inverting the coefficient matrix.
7x + 2y = 1
3x + y = 5
Chapter 1
,Elementary Linear Algebra 12e –7– Anton/Rorres
22. Solve the following matrix equation for X.
⎡ ⎤ ⎡ ⎤
1 2 3 2 2 3 0
⎢ ⎥ ⎢ ⎥
⎢0 1 4⎥ X = ⎢ 0 0 0 1⎥
⎣ ⎦ ⎣ ⎦
5 6 0 3 1 1 1
⎡ ⎤ ⎡ ⎤
1 0 0 3
⎢ ⎥ ⎢ ⎥
23. Given that A = ⎢
−1 ⎥
⎣0 2 1⎦ and b = ⎣1⎦,
⎢ ⎥ solve the system A2 x = b.
0 1 0 2
24. Find a nonzero solution to the following equation.
1 3
x = 3x
4 −3
25. Find the values of a, b, and c that make the following matrix symmetric.
⎡ ⎤
3 a 2−b
⎢ ⎥
⎢ 4 0 a + b⎥
⎣ ⎦
2 c 7
⎡ ⎤ ⎡ ⎤
3 4 3 1 −7 6
⎢ ⎥ ⎢ ⎥
26. Let A = ⎢
⎣ 0 0 6 ⎥, B = ⎢−4
⎦ ⎣ 5 0 ⎥, and AB = [cij ].
⎦
0 0 2 1 0 2
Find the diagonal entries c11 , c22 , and c33 .
27. Let the entries of a matrix A = [aij ] be defined as aij = 2i2 − i + j + g(j), where g is a
function of j. If A is a symmetric matrix, what is g(j)?
28. Prove that for any square matrix A, the matrix B = (A + AT ) is symmetric.
29. Find the domain and codomain of the transformation defined by
⎡ ⎤
x1
⎢ ⎥
5 7 6 0 ⎢ x2 ⎥
⎢
⎥
⎢ ⎥
1 0 −2 −2 ⎢ ⎥
⎣ x3 ⎦
x4
30. Find the standard matrix for the operator T : R2 → R2 defined by
3x1 + x2 = w1
4x2 = w2
Chapter 1
, Elementary Linear Algebra 12e –8– Anton/Rorres
31. Find the standard matrix for the transformation T defined by the formula
T (x1 , x2 , x3 ) = (x1 , −x3 , x2 − x1 , 3x2 + x3 )
32. Find the standard matrix A for the linear transformation T : R2 → R2 for which
2 3 −1 5
T = ,T =
−1 −4 1 2
33. Prove that the composition of two rotation operators about the origin of R2 is another
rotation about the origin.
34. Prove that if TA : R3 → R3 and TA (x) = 0 for every vector x in R3 , then A is the 3 × 3
zero matrix.
35. Write a balanced equation for the following chemical reaction.
C3 H8 + O2 → H2 O + CO2
36. Find the quadratic polynomial whose graph passes through the points 0, 3 , −1, 8 ,
and 1, 0 .
37. Use matrix inversion to find the production vector x that meets the demand d for the
consumption matrix C. ⎡ ⎤ ⎡ ⎤
0.1 0.3 0.2 18
⎢ ⎥ ⎢ ⎥
C=⎢ ⎥ ⎢ ⎥
⎣0.5 0.1 0.2⎦ ; d = ⎣40⎦
0.2 0.4 0.3 26
Chapter 1
